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Transcript
Foundations of Mathematics CaLc Math Linear Equations in a Nutshell fx = -0.5x-1 General Equation for a line: gx = 2x+1 8 hx = -0.5x-4 qx = -0.5x+4 y = mx + b 6 4 Slope: 2 Method 1: Counting the Squares rise slope m run -10 Example m= -5 5 -2 NOTE: -4 Method 2: The equation -6 y2 y1 slope m E.g. What’s the slope of a line that passes through (-2, -5) and (5, 2)? x2 x1 -8 NOTE: What if the equation isn’t in “y = mx + b” form? You should Example 1: 2x + 4y = 8 Example 2: 8x – y = 10 MFM2P Tuesday, June 27, 2017 Linear Equations in a nutshell page 1 of 5 8 Foundations of Mathematics fx = -0.5x-1 CaLc Math 6 g x = 2x+1 8 Special Slope properties: h x = -0.5x-4 4 q x = -0.5x+4 Parallel Lines 6 Perpendicular Lines 2 4 2 -5 5 -2 -10 10 -5 5 -2 -4 -4 -6 slope of line 3: m = slope of line 1: m = -6 -8 slope of line 2: m = slope of line 4: m = Conclusion: Conclusion: Example 1: Write the equation of the line that is perpendicular to y = ¾ x + 5 and has a y-int. of (0,1)? Example 2: What is the equation of a line is parallel to a line that passes through (2,8) and (4, 5) and has a y-intercept of (0, -15)? MFM2P Tuesday, June 27, 2017 -8 Linear Equations in a nutshell page 2 of 5 1 Foundations of Mathematics CaLc Math Graphing: There are 2 ways to graph a line: Method 1: Table of values. Example: Graph y = - 2x + 1 X Y = -2x + 1 10 8 6 4 2 -10 -5 5 10 5 10 -2 -4 -6 10 -8 Method 2: y = mx + b method 8 Graph y = - ¾ x – 2 6 Slope: 4 2 y-intercept: -10 -5 -2 -4 -6 -8 MFM2P Tuesday, June 27, 2017 Linear Equations in a nutshell page 3 of 5 Foundations of Mathematics CaLc Math Solutions to linear systems: In the algebra unit, we used elimination and substitution to solve for the (x, y). Another method is graphing, and where the two lines meet, it is the solution. Examples: Part 1 2 y x 2 3 1 y x1 3 Solution: (x, y) = ( Part 2 ** REARRANGE EQ’ns to y=mx+b! x – 2y +8 = 0 -x + 4y =24 , ) Solution: (x, y) = ( , ) 10 8 6 4 2 -10 -5 5 10 -2 -4 -6 -8 MFM2P Tuesday, June 27, 2017 Linear Equations in a nutshell page 4 of 5 Foundations of Mathematics CaLc Math Practice Questions: Question 1: Using the 2nd method of graphing lines. identify the b and then use the slope!) 2 a. y x 7 5 b. y 2 x 4 c. y x 2 3 Graph these lines (Hint: Remember to first 2 x5 3 e. y=-4x -3 f. –1x + 4y =8 (Rearrange this equation to get y =mx+b form) d. y Question 2: Graph these systems of equations to find the SOLUTION (where they intersect) a) y = 2x – 4 b) y = -1x + 3 y = 3x – 6 y=x+1 c) y = 2x + 1 y = - ½x + 1 d) y = 7x - 2 y = -x + 6 e) y = 2x + 1 y = 3x + 5 f) y = -x – 4 y=x Question 3: a. Write the equation of the line that is parallel to y = 3x + 6 and has a y-intercept of (0,-3). b. A line is perpendicular to y = 7x – 12 and passes through (0, 3). What is the equation? c. A line parallel to the line y = - ½ x + 9 and passes through (0, 19). What is the equation? d. A line is perpendicular to y = - ½ x + 9 and passes through (0, 19). What is the equation? e. There is a line is parallel to the line that passes through (9, 10) and (11, 12). It has a yintercept of (0, -20). What is the equation? f. There is a line that is perpendicular to the line that passes through (-3, -5) and (1, 2); the perpendicular line has b = 3. What is the equation of the line? Question 4: Rearrange the equation to create the y = mx + b form. Then identify your m & b a. y + 7x = 12 e. 4x – y = 9 b. x + 3y = 7 f. 8y – x = -4 c. y – 8 = 5x g. 3x + y – 21 d. x – 7y = 10 h. x – 7y = 0 Question 5: Graph these equations to find the SOLUTION of the linear system. (Rearrange these equations to the y=mx+b form before you graph them.) a. y = 3x - 5 b. 3x – y = 4 2x – 4y = 10 y=x+1 c. -2y = x – 8 2x – 4y = - 16 d. y = -2x - 6 -x – 3y = 13 e. 4x – 5 = y x – 4y = 5 f. y = -4x – 3 2x – y = 3 MFM2P Tuesday, June 27, 2017 Linear Equations in a nutshell page 5 of 5